Numerical Methods To Solve Initial Value Problems An

Numerical Methods To Solve Initial Value Problems An Over View of Runge-Kutta Fehlberg and Dormand Prince Methods. William Mize

Quick Refresher �

A Problem �

Some Quick Ground work �First Start with Taylor Series Approximations �Then Move onto Runge-Kutta Methods for Approximations �Lastly onto Runge-Kutta Fehlberg and Dormand Prince Methods for Approximation and keeping control of error

How these Methods Work �All of the Methods will be using a step size method. �Error is determined by the size of step, order, and method used. �When actually calculating these, almost always done via computer.

Taylor Series Methods(Brief) �

Runge-Kutta Methods �Named After Carl Runge and Wilhelm Kutta �What they do? �Do the same Job as Taylor Series Method, but without the analytic differentiation. �Just like Taylor Series with higher and higher order methods. �Runge-Kutta Method of Order 4 Well accepted classically used algorithm.

Runge-Kutta of Order 2 �

Runge-Kutta of Order 4 �

So What's next? �Already Viable Numerical Solution established what's the next step? �We want to control our Error and Step size at each step. �These methods are called adaptive. �Why? �Cost Less �Keep within Tolerance �Also look for More efficient ways of doing these things. � 10 Function Evaluation for RK 4 and RK 5 �Just 6 for RKF 4(5)

Runge-Kutta Fehlberg �

Next Step to find These Coefficients

Further Deriving �

More and more… �

Comparison(Problem)

Comparisons of Methods

Dormand Prince Methods

Visual Comparison of Methods

Conclusion �Taylor’s method uses derivatives to solve ODE �RK uses only a combination of specific function evaluations instead of derivatives to approximate solution of the ODE �RKF is beneficial because you can control your step size so you have your global error within a predetermined tolerance �RK 4 and RK 5 uses 10 function evaluations vs RKF just 6 �Runge-Kutta Fehlberg is widely accepted and used commercially(Matlab, Mathematica, maple, etc)

Sources �Numerical Mathematics and Computing. Sixth Edition; Ward Cheny, David Kincaid �Low-Order classical Runge-Kutta Formulas with Step. Size Control and their Application to some heat transfer problems. By Erwin Fehlberg(1969) �A family of embedded Runge-Kutta Formulae. By Dormand Prince(1980)